Titanium-diffused lithium niobate $\left(\text{Ti}:{\text{LiNbO}}_3\right)$ waveguides are widely used in current fiber optic networks because of their high-speed, electro-optic modulation, and low-loss integration with standard single-mode fibers. However, they cannot achieve small ring resonators owing to their lack of a high core-to-cladding index contrast. To overcome this challenge, we vertically integrate an emerging chalcogenide glass waveguide technology on well-established $\text{Ti}:{\text{LiNbO}}_3$ waveguides. We present, to our knowledge, the first arsenic trisulfide $\left({\text{As}}_2{\text{S}}_3\right)$ race-track ring resonator with a $290.8\mu \text{m}$ bend radius in an all-pass filter configuration, integrated on a $\text{Ti}:{\text{LiNbO}}_3$ waveguide. Vertical coupling is achieved using a unique taper design. Experimental results are shown for 10.6% coupling, 2.08 dB roundtrip loss, and a 25.4 GHz free-spectral range.